Prove that the following function $f(X)=XX^{t}$ is differentiable

Question

Let $M(n \times n) \simeq \mathbb{R}^{n^2}$ and $f:\mathbb{R}^{n^2} \rightarrow \mathbb{R}^{n^2}$ defined by $f(X)=XX^{t}$.

Prove que $f$ is differentiable and find $Df(X).A$.

I'm having a lot of difficulties in this exercise. I used the definition of Differentiability,, that is, $\dfrac{f(a+h)-f(a)-B.h}{||h||}\rightarrow 0$ as $h \rightarrow 0$. I only got

$\dfrac{(A+H)(A+H)^t - AA^t - BH}{||H||}=\dfrac{AH^t + HA^t+ HH^t - BH}{||H||}$.

Does anyone have any idea how to proceed? Or a better idea?